Canousmic temperaments: Difference between revisions
m +link to marthirds |
+proposed temperament: superlimmal (80 & 311) |
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* ''[[kleischismic]]'', {32805/32768, 1562500/1492992} → [[Schismatic family #Kleischismic]] | * ''[[kleischismic]]'', {32805/32768, 1562500/1492992} → [[Schismatic family #Kleischismic]] | ||
Discussed below are satin and semiluna. | Discussed below are satin, superlimmal and semiluna. | ||
= Satin = | == Satin == | ||
== 5-limit == | === 5-limit === | ||
Subgroup: 2.3.5 | Subgroup: 2.3.5 | ||
| Line 30: | Line 30: | ||
[[Badness]]: 2.8530 | [[Badness]]: 2.8530 | ||
== 7-limit == | === 7-limit === | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 44: | Line 44: | ||
[[Badness]]: 0.1972 | [[Badness]]: 0.1972 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 58: | Line 58: | ||
Badness: 0.0580 | Badness: 0.0580 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 72: | Line 72: | ||
Badness: 0.0303 | Badness: 0.0303 | ||
== 17-limit == | === 17-limit === | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
| Line 86: | Line 86: | ||
Badness: 0.0200 | Badness: 0.0200 | ||
== 19-limit == | === 19-limit === | ||
Subgroup: 2.3.5.7.11.13.17.19 | Subgroup: 2.3.5.7.11.13.17.19 | ||
| Line 100: | Line 100: | ||
Badness: 0.0145 | Badness: 0.0145 | ||
== 23-limit == | === 23-limit === | ||
Subgroup: 2.3.5.7.11.13.17.19.23 | Subgroup: 2.3.5.7.11.13.17.19.23 | ||
| Line 114: | Line 114: | ||
Badness: 0.0122 | Badness: 0.0122 | ||
= Semiluna = | == Superlimmal == | ||
The superlimmal temperament uses an ever slightly sharpened [[27/25|large limma]] as the generator, nine exceed the octave by [[126/125]]. It can be described as 80 & 311. It gets all the primes up to 29 reasonably covered, but still acceptible just as a 13-limit microtemperament, judging from its comma basis. The 80-tone [[MOS scale]] is presumably the place to start, and a 151-tone MOS is possible. | |||
Subgroup: 2.3.5.7 | |||
[[Comma list]]: 4802000/4782969, 52734375/52706752 | |||
[[Mapping]]: [{{val| 1 8 12 18 }}, {{val| 0 -57 -86 -135 }}] | |||
{{Multival|legend=1| 57 86 135 3 53 72 }} | |||
[[POTE generator]]: ~27/25 = 135.0464 | |||
{{Val list|legend=1| 80, 231, 311, 1324b, 1635b }} | |||
[[Badness]]: 0.2524 | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 3025/3024, 4000/3993, 1479016/1476225 | |||
Mapping: [{{val| 1 8 12 18 11 }}, {{val| 0 -57 -86 -135 -67 }}] | |||
POTE generator: ~27/25 = 135.0455 | |||
Vals: {{val list| 80, 231, 311, 1013e, 1324be }} | |||
Badness: 0.06067 | |||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 3025/3024, 4000/3993, 4225/4224, 4459/4455 | |||
Mapping: [{{val| 1 8 12 18 11 1 }}, {{val| 0 -57 -86 -135 -67 24 }}] | |||
POTE generator: ~27/25 = 135.0446 | |||
Vals: {{val list| 80, 231, 311, 702, 1013e }} | |||
Badness: 0.03902 | |||
=== 17-limit === | |||
Subgroup: 2.3.5.7.11.13.17 | |||
Comma list: 595/594, 1275/1274, 2500/2499, 3025/3024, 4225/4224 | |||
Mapping: [{{val| 1 8 12 18 11 1 6 }}, {{val| 0 -57 -86 -135 -67 24 -17 }}] | |||
POTE generator: ~27/25 = 135.0462 | |||
Vals: {{val list| 80, 231, 311 }} | |||
Badness: 0.03008 | |||
=== 19-limit === | |||
Subgroup: 2.3.5.7.11.13.17.19 | |||
Comma list: 595/594, 969/968, 1275/1274, 1445/1444, 1729/1728, 2500/2499 | |||
Mapping: [{{val| 1 8 12 18 11 1 6 11 }}, {{val| 0 -57 -86 -135 -67 24 -17 -60 }}] | |||
POTE generator: ~27/25 = 135.0464 | |||
Vals: {{val list| 80, 231, 311 }} | |||
Badness: 0.02046 | |||
=== 23-limit === | |||
Subgroup: 2.3.5.7.11.13.17.19.23 | |||
Comma list: 595/594, 760/759, 969/968, 1105/1104, 1275/1274, 1445/1444, 1496/1495 | |||
Mapping: [{{val| 1 8 12 18 11 1 6 11 7 }}, {{val| 0 -57 -86 -135 -67 24 -17 -60 -22 }}] | |||
POTE generator: ~27/25 = 135.0458 | |||
Vals: {{val list| 80, 231, 311 }} | |||
Badness: 0.01615 | |||
=== 29-limit === | |||
Subgroup: 2.3.5.7.11.13.17.19.23.29 | |||
Comma list: 595/594, 760/759, 784/783, 969/968, 1045/1044, 1105/1104, 1275/1274, 1496/1495 | |||
Mapping: [{{val| 1 8 12 18 11 1 6 11 7 16 }}, {{val| 0 -57 -86 -135 -67 24 -17 -60 -22 -99 }}] | |||
POTE generator: ~27/25 = 135.0460 | |||
Vals: {{val list| 80, 231, 311 }} | |||
Badness: 0.01305 | |||
== Semiluna == | |||
{{see also|Luna family #Semiluna}} | {{see also|Luna family #Semiluna}} | ||
| Line 129: | Line 224: | ||
[[Badness]]: 0.1922 | [[Badness]]: 0.1922 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Revision as of 06:39, 12 June 2021
These are rank-2 temperaments that temper out the canousma, 4802000/4782969 = [4 -14 3 4⟩. For the rank-3 temperament, see Canou family.
Note that 4802000/4782969 = 2×(10/9)3/(9/7)4, these intervals tend to have lower complexity.
Temperaments not dicussed here include:
- godzilla, {49/48, 81/80} → Meantone family #Godzilla
- betic, {225/224, 1071875/1062882} → Sycamore family #Betic
- pentorwell, {1728/1715, 179200/177147} → Orwellismic temperaments #Pentorwell
- amicable, {2401/2400, 1600000/1594323} → Breedsmic temperaments #Amicable
- parakleismic, {3136/3125, 4375/4374} → Ragismic microtemperaments #Parakleismic
- septiquarter, {5120/5103, 420175/419904} → Hemifamity temperaments #Septiquarter
- marthirds, {15625/15552, 2460375/2458624} → Kleismic family #Marthirds
- kleischismic, {32805/32768, 1562500/1492992} → Schismatic family #Kleischismic
Discussed below are satin, superlimmal and semiluna.
Satin
5-limit
Subgroup: 2.3.5
Comma list: [104 -70 3⟩
Mapping: [⟨1 2 12], ⟨0 -3 -70]]
POTE generator: ~[-34 23 -1⟩ = 165.907
Badness: 2.8530
7-limit
Subgroup: 2.3.5.7
Comma list: 2100875/2097152, 4802000/4782969
Mapping: [⟨1 2 12 -3], ⟨0 -3 -70 42]]
POTE generator: ~8575/7776 = 165.913
Badness: 0.1972
11-limit
Subgroup: 2.3.5.7.11
Comma list: 4000/3993, 19712/19683, 41503/41472
Mapping: [⟨1 2 12 -3 13], ⟨0 -3 -70 42 -69]]
POTE generator: ~11/10 = 165.915
Badness: 0.0580
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 1575/1573, 2080/2079, 4096/4095, 13720/13689
Mapping: [⟨1 2 12 -3 13 -1], ⟨0 -3 -70 42 -69 34]]
POTE generator: ~11/10 = 165.914
Badness: 0.0303
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 595/594, 833/832, 1156/1155, 1575/1573, 4096/4095
Mapping: [⟨1 2 12 -3 13 -1 11], ⟨0 -3 -70 42 -69 34 -50]]
POTE generator: ~11/10 = 165.913
Badness: 0.0200
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 595/594, 833/832, 969/968, 1156/1155, 1216/1215, 1575/1573
Mapping: [⟨1 2 12 -3 13 -1 11 16], ⟨0 -3 -70 42 -69 34 -50 -85]]
POTE generator: ~11/10 = 165.913
Badness: 0.0145
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 595/594, 760/759, 833/832, 875/874, 969/968, 1105/1104, 1156/1155
Mapping: [⟨1 2 12 -3 13 -1 11 16 16], ⟨0 -3 -70 42 -69 34 -50 -85 -83]]
POTE generator: ~11/10 = 165.914
Badness: 0.0122
Superlimmal
The superlimmal temperament uses an ever slightly sharpened large limma as the generator, nine exceed the octave by 126/125. It can be described as 80 & 311. It gets all the primes up to 29 reasonably covered, but still acceptible just as a 13-limit microtemperament, judging from its comma basis. The 80-tone MOS scale is presumably the place to start, and a 151-tone MOS is possible.
Subgroup: 2.3.5.7
Comma list: 4802000/4782969, 52734375/52706752
Mapping: [⟨1 8 12 18], ⟨0 -57 -86 -135]]
Wedgie: ⟨⟨ 57 86 135 3 53 72 ]]
POTE generator: ~27/25 = 135.0464
Badness: 0.2524
11-limit
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 4000/3993, 1479016/1476225
Mapping: [⟨1 8 12 18 11], ⟨0 -57 -86 -135 -67]]
POTE generator: ~27/25 = 135.0455
Vals: Template:Val list
Badness: 0.06067
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 3025/3024, 4000/3993, 4225/4224, 4459/4455
Mapping: [⟨1 8 12 18 11 1], ⟨0 -57 -86 -135 -67 24]]
POTE generator: ~27/25 = 135.0446
Vals: Template:Val list
Badness: 0.03902
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 595/594, 1275/1274, 2500/2499, 3025/3024, 4225/4224
Mapping: [⟨1 8 12 18 11 1 6], ⟨0 -57 -86 -135 -67 24 -17]]
POTE generator: ~27/25 = 135.0462
Vals: Template:Val list
Badness: 0.03008
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 595/594, 969/968, 1275/1274, 1445/1444, 1729/1728, 2500/2499
Mapping: [⟨1 8 12 18 11 1 6 11], ⟨0 -57 -86 -135 -67 24 -17 -60]]
POTE generator: ~27/25 = 135.0464
Vals: Template:Val list
Badness: 0.02046
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 595/594, 760/759, 969/968, 1105/1104, 1275/1274, 1445/1444, 1496/1495
Mapping: [⟨1 8 12 18 11 1 6 11 7], ⟨0 -57 -86 -135 -67 24 -17 -60 -22]]
POTE generator: ~27/25 = 135.0458
Vals: Template:Val list
Badness: 0.01615
29-limit
Subgroup: 2.3.5.7.11.13.17.19.23.29
Comma list: 595/594, 760/759, 784/783, 969/968, 1045/1044, 1105/1104, 1275/1274, 1496/1495
Mapping: [⟨1 8 12 18 11 1 6 11 7 16], ⟨0 -57 -86 -135 -67 24 -17 -60 -22 -99]]
POTE generator: ~27/25 = 135.0460
Vals: Template:Val list
Badness: 0.01305
Semiluna
Subgroup: 2.3.5.7
Comma list: 4802000/4782969, 95703125/95551488
Mapping: [⟨2 8 4 23], ⟨0 -15 2 -54]]
POTE generator: ~2187/1960 = 193.1725
Badness: 0.1922
11-limit
Subgroup: 2.3.5.7.11
Comma list: 5632/5625, 9801/9800, 14641/14580
Mapping: [⟨2 8 4 23 14], ⟨0 -15 2 -54 -22]]
POTE generator: ~121/108 = 193.1732
Badness: 0.0678